Calculate the height and volume of a solid composed of a cone, cylinder, and hemisphere.

Here are the calculations for the solid:

Given:
Diameter $D = 10 \text{ cm}$
Radius $r = \frac{D}{2} = \frac{10}{2} = 5 \text{ cm}$
Height of cylindrical part $h_{cyl} = 15 \text{ cm}$
Slant height of conical part $l = 13 \text{ cm}$
$\pi = 3.142$

a) Height of the solid

Step 1: Calculate the height of the conical part ($h_{cone}$).
Using the Pythagorean theorem for the cone: $l^2 = r^2 + h_{cone}^2$.

$$13^2 = 5^2 + h_{cone}^2$$ $$169 = 25 + h_{cone}^2$$ $$h_{cone}^2 = 169 - 25$$ $$h_{cone}^2 = 144$$ $$h_{cone} = \sqrt{144} = 12 \text{ cm}$$

Step 2: Determine the height of the hemispherical part ($h_{hemi}$).
The height of the hemispherical part is equal to its radius.

$$h_{hemi} = r = 5 \text{ cm}$$

Step 3: Calculate the total height of the solid.
The total height is the sum of the heights of the conical, cylindrical, and hemispherical parts.

$$H_{total} = h_{cone} + h_{cyl} + h_{hemi}$$ $$H_{total} = 12 \text{ cm} + 15 \text{ cm} + 5 \text{ cm}$$ $$H_{total} = 32 \text{ cm}$$

The height of the solid is $\boxed{\text{32.0 cm}}$.

b) Volume of the solid, correct to 1 decimal place

Step 1: Calculate the volume of the conical part ($V_{cone}$).

$$V_{cone} = \frac{1}{3} \pi r^2 h_{cone}$$ $$V_{cone} = \frac{1}{3} \times 3.142 \times (5 \text{ cm})^2 \times 12 \text{ cm}$$ $$V_{cone} = \frac{1}{3} \times 3.142 \times 25 \times 12 \text{ cm}^3$$ $$V_{cone} = 3.142 \times 25 \times 4 \text{ cm}^3$$ $$V_{cone} = 314.2 \text{ cm}^3$$

Step 2: Calculate the volume of the cylindrical part ($V_{cyl}$).

$$V_{cyl} = \pi r^2 h_{cyl}$$ $$V_{cyl} = 3.142 \times (5 \text{ cm})^2 \times 15 \text{ cm}$$ $$V_{cyl} = 3.142 \times 25 \times 15 \text{ cm}^3$$ $$V_{cyl} = 1178.25 \text{ cm}^3$$

Step 3: Calculate the volume of the hemispherical part ($V_{hemi}$).

$$V_{hemi} = \frac{2}{3} \pi r^3$$ $$V_{hemi} = \frac{2}{3} \times 3.142 \times (5 \text{ cm})^3$$ $$V_{hemi} = \frac{2}{3} \times 3.142 \times 125 \text{ cm}^3$$ $$V_{hemi} = \frac{785.5}{3} \text{ cm}^3$$ $$V_{hemi} \approx 261.8333 \text{ cm}^3$$

Step 4: Calculate the total volume of the solid.

$$V_{total} = V_{cone} + V_{cyl} + V_{hemi}$$ $$V_{total} = 314.2 \text{ cm}^3 + 1178.25 \text{ cm}^3 + 261.8333 \text{ cm}^3$$ $$V_{total} = 1754.2833 \text{ cm}^3$$

Rounding to 1 decimal place:

$$V_{total} = 1754.3 \text{ cm}^3$$

The volume of the solid is $\boxed{\text{1754.3 cm}^3}$.

c) Surface area of the solid, correct to 1 decimal place

Step 1: Calculate the curved surface area of the conical part ($CSA_{cone}$).

$$CSA_{cone} = \pi r l$$ $$CSA_{cone} = 3.142 \times 5 \text{ cm} \times 13 \text{ cm}$$ $$CSA_{cone} = 3.142 \times 65 \text{ cm}^2$$ $$CSA_{cone} = 204.23 \text{ cm}^2$$

Step 2: Calculate the curved surface area of the cylindrical part ($CSA_{cyl}$).

$$CSA_{cyl} = 2 \pi r h_{cyl}$$ $$$$